Building a roof or cutting stringers for a stairway requires finding angles, marking and cutting them correctly. In carpentry, you usually do away with trigonometric functions to find angles. Instead, you simply use two legs of a triangle to measure and mark the angle. Call one leg the rise, designating the height of the triangle, and the other leg the run, or the length of the triangle. In some cases however, you still have to use basic trigonometry to determine an angle.
Reduce the length of the run and the height of the rise by the same factor so they can be measured with a framing square. For example, the angle made by a run of 8 feet and a rise of 12 feet makes the same angle as a run of 8 inches and 12 inches.
Place the framing square on the lumber and position it so that both measurements align with the edge of the lumber piece. In the example, place the 8-inch mark on one leg against the edge and the 12-inch mark on the other leg against the same edge.
Use the edges of the square to mark the rise and the run of the triangle. The resulting cuts reflect two of the three angles on the triangle. The third angle is 90 degrees. One use of this method marks the cut line on stair stringers after calculating the rise and run of the steps.
Place the square on the board with the T-shaped edge against the board edge. Move the square so that the pivot point at the right angle of the triangle is aligned with the board where the cut is to be made.
Draw a line across the board using the edge between the pivot point and the diagonal side of the square as a guide. This line is perfectly perpendicular to the edge of the board.
Rotate the square around the pivot point until the desired angle on the diagonal side of the square is aligned with the same edge of the board that the pivot point rests against. Mark a line across the board using the side of the square between the pivot point and the diagonal side as a guide. This line marks the desired angle with the first line.
Calculate the length of the diagonal leg of a right triangle. Square both lengths of the other two sides, add them together and take the square root. The result is the length of the diagonal leg. The length of the diagonal leg is used to find the angles between the other two legs and the diagonal leg.
Calculate the angle between the horizontal and diagonal legs of a right triangle. Divide the length of the horizontal leg by the length of the diagonal leg using the calculator. Press the inverse button, then press the cosine button to display the angle between the horizontal and diagonal legs.
Calculate the angle between the vertical and diagonal legs of a right triangle. Divide the length of the vertical leg by the length of the diagonal leg using the calculator. Press the inverse button, then press the cosine button to display the angle between the vertical and diagonal legs.
There are many carpentry applications of these three methods of finding angles. From roof slopes to stairways to architectural details, the application of these three methods is easily applied with a little thought and planning. Right triangles are the basis for many carpentry and construction methods. Any triangle that is not a right triangle turns into a pair of right triangles with the addition of a single drawn line.